SOLUTION: A goat is tied outside a triangular fenced garden at point A. The sides of the fence are AB = 8 m, BC = 9 m, and CA = 12 m. If the rope with which the goat is tied is 14 m long, fi

The figure below is drawn to scale.  The red line is the rope, with 
the goat pulling it tight.  The circle has center A and radius 14 m.



The goat can graze anywhere in the circle except in the triangle and
in the area marked with an X in the figure below:



The desired area consists of three sectors.
1. The big sector whose center is A and has radius 14 m. Its
central angle is 360°-A 
2. The smallest sector whose center is C and has radius 2 m. Its
central angle is 180°-C.  
3. The next to smallest sector whose center is B and has radius 6 m.  
We will need the exterior angle at B to calculate its area. That will
be the sum of the two remote interior angles, A+C

We find all three angles of the triangle using the law of cosines.



The formula for the area of a sector is 



The area of the big sector: Its radius is r = 14 m. 
Its central angle is θ = 360°-A = 360°-48.58881136° = 311.4111886°.


The area of the smallest sector: Its radius is r = 2 m. 
Its central angle is θ = 180°-C = 180°-41.80907919° = 138.1909208°.


The area of the next to smallest sector: Its radius is r = 6 m. 
Its central angle is θ = A+C = 48.58881136° + 41.80907919° = 90.39789055°


Finally we add the three sectors together to get the entire grazing area:

532.6447557 + 4.823773129 + 28.39933489 = 565.8678637 m².

The correct answer is B.

Edwin